This research aims to assemble two methodical spectral Legendre's derivative algorithms to
numerically attack the Lane-Emden, Bratu's, and singularly perturbed type equations. We …

The usual classical polynomials-based spectral Galerkin and Petrov–Galerkin methods
enjoy high-order accuracy for problems with smooth solutions. However, their accuracy and …

Herein, we propose new efficient spectral algorithms for handling the fractional diffusion
wave equation (FDWE) and fractional diffusion wave equation with damping (FDWED). In …

This paper develops and analyses a finite difference/spectral-Galerkin scheme for the
nonlinear fractional Schrödinger equations with Riesz space-and Caputo time-fractional …

A new numerical scheme based on the tau spectral method for solving the linear hyperbolic
telegraph type equation is presented and implemented. The derivation of this scheme is …

In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre
spectral scheme for the nonlinear Riesz-space and Caputo-time fractional reaction–diffusion …

One of the issues in numerical solution analysis is the non-linear distributed-order fractional
Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We …

The current manuscript introduces a novel numerical treatment for multi-term fractional
differential equations with variable coefficients. The spectral Galerkin approach is developed …

An algorithm based on a class of the Chebyshev polynomials family called the fifth-kind
Chebyshev polynomials (FCPs) is introduced to solve the multi-term variable-order time …

We present a class of orthogonal functions on infinite domain based on Jacobi polynomials.
These functions are generated by applying a tanh transformation to Jacobi polynomials. We …